OtjPLdZddlmZddlZddlmZmZddlmZddl m Z ddl m Z ddl mZdd lmZdd lmZmZdd lmZmZmZmZdd lmZdd lmZmZmZmZddl m!Z!ddl"m#Z#GddeZ$ d dZ%dZ&d!dZ'd!dZ(d!dZ)d!dZ*d!dZ+d!dZ,e#dddddZ-dS)"z Compute Galois groups of polynomials. We use algorithms from [1], with some modifications to use lookup tables for resolvents. References ========== .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory*. ) defaultdictN)Dummysymbols) is_squareZZ) dup_random)dup_eval)dup_discriminant)dup_factor_listdup_irreducible_p)GaloisGroupExceptionget_resolvent_by_lookupdefine_resolvents Resolvent) coeff_search)Polypoly_from_exprPolificationFailedComputationFailed) dup_sqf_p)publicceZdZdS)MaxTriesExceptionN)__name__ __module__ __qualname__c/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/sympy/polys/numberfields/galoisgroups.pyrr#sCrr Tc~td}|}|t}||j|rid}d}fd} t |D]K} |r| |} t | } td| D} || z|kr0|dkr |dz }|dz }n|dz}| |} t | } tdgd| Dz}nFt| d zdz|}tj d|dz }t|| |t}t||j}t|||z |}|j|vr2t!|jtr||fcSMt$) a Given a univariate, monic, irreducible polynomial over the integers, find another such polynomial defining the same number field. Explanation =========== See Alg 6.3.4 of [1]. Parameters ========== T : Poly The given polynomial max_coeff : int When choosing a transformation as part of the process, keep the coeffs between plus and minus this. max_tries : int Consider at most this many transformations. history : set, None, optional (default=None) Pass a set of ``Poly.rep``'s in order to prevent any of these polynomials from being returned as the polynomial ``U`` i.e. the transformation of the given polynomial *T*. The given poly *T* will automatically be added to this set, before we try to find a new one. fixed_order : bool, default True If ``True``, work through candidate transformations A(x) in a fixed order, from small coeffs to large, resulting in deterministic behavior. If ``False``, the A(x) are chosen randomly, while still working our way up from small coefficients to larger ones. Returns ======= Pair ``(A, U)`` ``A`` and ``U`` are ``Poly``, ``A`` is the transformation, and ``U`` is the transformed polynomial that defines the same number field as *T*. The polynomial ``A`` maps the roots of *T* to the roots of ``U``. Raises ====== MaxTriesException if could not find a polynomial before exceeding *max_tries*. XNcZ|t|d}||<|S)N)getr)degreegencoeff_generatorss r get_coeff_generatorz9tschirnhausen_transformation..get_coeff_generatorcs2""6<+B+BCC#&  rc34K|]}t|VdS)N)abs.0cs r z/tschirnhausen_transformation..xs(++qCFF++++++rr(c,g|]}t|Srrr0s r z0tschirnhausen_transformation..s111Q2a55111r)rr*setaddreprangenextmaxrminrandomrandintr rr+ resultantrto_listr)T max_coeff max_trieshistory fixed_orderr$n deg_coeff_sumcurrent_degreer-ir+coeffsmaCdAUr,s @r tschirnhausen_transformationrR'sb c A  A%% KK  9  &&  ) &%n55C#YYF++F+++++A!M11!Q&&!Q&M%2Q%6NN"a'N)).99cA11&1111AA AqD1Hi((Aq!a%((A1qb!R((A AENN QU##Q ' ' 5  Iaemmoor$B$B a4KKK rct|tr|nt|t}t |S)z?Convenience to check if a Poly or dup has square discriminant. ) isinstancer discriminantr rr)rBrOs r has_square_discrVs<&q$//L5Ea5L5LA Q<<rFcPddlm}t|r |jdfn|jdfS)z~ Compute the Galois group of a polynomial of degree 3. Explanation =========== Uses Prop 6.3.5 of [1]. r)S3TransitiveSubgroupsTF)sympy.combinatorics.galoisrXrVA3S3)rBrD randomizerXs r _galois_group_degree_3r]sEA@@@@@0?0B0B 3 " %t , ,'*E24rc ddlm}ddlm}t d}|d|dz|d|dzz}|d|ddd|dddg}t |||}|d|ddzz|d|ddzzz|d|ddzzz|d|ddzzz} |d|dddg} t } t|D]3} | dkrt||| | \} }| |d \}} }t|tsQt|}||r |j d fn|jd fcS|r |jd fcS||| t#||d }fd| D}t |||}| |\}} } t%|t}|dkrt'|r |jd fcS|jd fcSt,)z Compute the Galois group of a polynomial of degree 4. Explanation =========== Follows Alg 6.3.7 of [1], using a pure root approximation approach. r PermutationS4TransitiveSubgroupsz X0 X1 X2 X3r&r(r%rDrErFT)find_integer_rootNF simultaneousc g|] }|zz Srrr1tausigmas r r5z6_galois_group_degree_4_root_approx..! 0 0 0#eCio 0 0 0r) sympy.combinatorics.permutationsr`rYrbrrr7r:rR eval_for_polyrrrVA4S4Vsubszipr rC4D4r)rBrDr\r`rbr$F1s1R1F2_pres2_prerErJ_R_dupi0sq_discF2s2R2rOrjs @r "_galois_group_degree_4_root_approxrs=<<<<<@@@@@@ A 1adQqT!A$Y B A Aq! Aq! B 2q"  B qT!A$'\AaD1qL (1Q4!a< 7!A$qtQw, FF A Aq!F eeG 9  .5.5 q55/Y8?@IMKKKDAq''T'BB q"##  "!$$ :9@;*-t44/2E: < < <  3)+T2 2 2 2 2[[Qa))[ = = 0 0 0 0 0 0 0 r1b ! !&&q)) q! UB ' ' 66  Q<< 5),e4 4 4 4),e4 4 4 4 rcddlm}t}t|D]@}t |d}t |t rnt|||| \}}Att|t }ttd|dDg} | dgkr!t|r |j dfn|jdfS| gd kr |jdfS| gd kr |jdfS| d d gksJ|jdfS) z Compute the Galois group of a polynomial of degree 4. Explanation =========== Based on Alg 6.3.6 of [1], but uses resolvent coeff lookup. rrarcc@g|]\}}t|dz g|zS)r(len)r1res r r5z1_galois_group_degree_4_lookup..s:!QQ! qrr(TFr(r()r&r&r&r&r)rYrbr7r:rrrrRrr sortedsumrVrnrorsrprt) rBrDr\rbrErJr{rzflLs r _galois_group_degree_4_lookuprsoA@@@@@eeG 9    '1-- UB    E+A4;.krkr)rYrrlr`rras_exprrr7r:rRrrrrVr A5S5M20 round_roots_to_integers_for_polyitemsr rqrrrmr rC5D5r)rBrDr\rr`X5resF51rzs51R51rEreached_second_stagerJR51_dupr} rounded_rootspermutation_indexcandidate_rootr$rxryr|r~rrr{rOrjs @r _galois_group_degree_5_hybridr)sAA@@@@@<<<<<< ! " "B   Cf+KCC #+r C CS ! !CeeG  9  6464 q55/Y8?@IMKKKDAq*!Q//"%%   $ :%a((G "-- @@@@ :-159999 $<.s5!Rc"gg r)r(r&r%r%Fr&rTrr6)r(r(r(r%r)#rYrr7r:rrrrRrr rlistrappendrrrrVC6D6G18G36mS4pA4xC2S4xC2rnS4mPSL2F5PGL2F5r[r A6S6G36pG72)rBrDr\rrErJr{rzrfactors_by_degrr T_has_sq_discf1f2 any_squares r _galois_group_degree_6_lookuprsA@@@@@eeG 9    '1-- UB    E+A4;*0%88/5u= ? iii4A8&)400+/7 9 q!f8E;&-t44+2E: < lll  %(%00 8888eeG 9    '1-- UB    E+A4;>> from sympy import galois_group >>> from sympy.abc import x >>> f = x**4 + 1 >>> G, alt = galois_group(f) >>> print(G) PermutationGroup([ (0 1)(2 3), (0 2)(1 3)]) The group is returned along with a boolean, indicating whether it is contained in the alternating group $A_n$, where $n$ is the degree of *T*. Along with other group properties, this can help determine which group it is: >>> alt True >>> G.order() 4 Alternatively, the group can be returned by name: >>> G_name, _ = galois_group(f, by_name=True) >>> print(G_name) S4TransitiveSubgroups.V The group itself can then be obtained by calling the name's ``get_perm_group()`` method: >>> G_name.get_perm_group() PermutationGroup([ (0 1)(2 3), (0 2)(1 3)]) Group names are values of the enum classes :py:class:`sympy.combinatorics.galois.S1TransitiveSubgroups`, :py:class:`sympy.combinatorics.galois.S2TransitiveSubgroups`, etc. Parameters ========== f : Expr Irreducible polynomial over :ref:`ZZ` or :ref:`QQ`, whose Galois group is to be determined. gens : optional list of symbols For converting *f* to Poly, and will be passed on to the :py:func:`~.poly_from_expr` function. by_name : bool, default False If ``True``, the Galois group will be returned by name. Otherwise it will be returned as a :py:class:`~.PermutationGroup`. max_tries : int, default 30 Make at most this many attempts in those steps that involve generating Tschirnhausen transformations. randomize : bool, default False If ``True``, then use random coefficients when generating Tschirnhausen transformations. Otherwise try transformations in a fixed order. Both approaches start with small coefficients and degrees and work upward. args : optional For converting *f* to Poly, and will be passed on to the :py:func:`~.poly_from_expr` function. Returns ======= Pair ``(G, alt)`` The first element ``G`` indicates the Galois group. It is an instance of one of the :py:class:`sympy.combinatorics.galois.S1TransitiveSubgroups` :py:class:`sympy.combinatorics.galois.S2TransitiveSubgroups`, etc. enum classes if *by_name* was ``True``, and a :py:class:`~.PermutationGroup` if ``False``. The second element is a boolean, saying whether the group is contained in the alternating group $A_n$ ($n$ the degree of *T*). Raises ====== ValueError if *f* is of an unsupported degree. MaxTriesException if could not complete before exceeding *max_tries* in those steps that involve generating Tschirnhausen transformations. See Also ======== .Poly.galois_group galois_groupr(Nr)rrrr) frrDr\gensargsFoptexcs r rrsD :2D :2D81D111D1133 88837778 >>'Y$-  / //s >9>)r!r"NT)r"F).__doc__ collectionsrr>sympy.core.symbolrrsympy.ntheory.primetestrsympy.polys.domainsrsympy.polys.densebasicr sympy.polys.densetoolsr sympy.polys.euclidtoolsr sympy.polys.factortoolsr r *sympy.polys.numberfields.galois_resolventsrrrr"sympy.polys.numberfields.utilitiesrsympy.polys.polytoolsrrrrsympy.polys.sqfreetoolsrsympy.utilitiesrrrRrVr]rrrrrrrrr rs  $##### ,,,,,,,,------""""""------++++++444444FFFFFFFF<;;;;;JJJJJJJJJJJJ------"""""",IM-1hhhhV 4 4 4 4TTTTn(-(-(-(-VNNNNb*0*0*0*0ZZ9Z9Z9Z9z#(B%j/j/j/j/j/j/j/r